Unramified Kummer extensions of prime power degree
Let \(R\) be a complete local domain with quotient field \(K\) of characteristic 0 and residue field \(k\) of characteristic \(p\), containing \(\zeta\), a primitive \(p^n\)-th root of unity. The author determines the Harrison group \(\mathrm{Gal}(R,G)\) of Galois extensions of \(R\) with group \(G\) cyclic of order \(p^n\). Group \(\mathrm{Gal}(K,G)\cong K^*/K^{*p^n}\) by Kummer theory; also \(\mathrm{Gal}(R,G)\cong \mathrm{Gal}(k,G)\), which is well-known by Artin-Schreier theory. The author's objective is to integrate the two theories by determining \(E_n(R)=\{x\in R^*\mid S_x=R[T]/(T^{p^n}- x)\) is embedded as \(G\)-module and \(R\)-algebra in a \(G\)-Galois extension of \(R\, \}\). Such an integration was achieved by \textit{H. Hasse} [J. Reine Angew. Math. 176, 174--183 (1936; Zbl 0016.05204)] using the Artin-Hasse exponential map. The paper under review obtains a description of \(E_n(R)\) as elements satisfying a certain congruence condition multiplied by values (at elements of \(R\)) of a special set of \(n\) polynomials with coefficients in \(\mathbb Z_p[\zeta]\); the polynomials are defined with the aid of the Artin-Hasse map. The description of \(E_n(R)\) for \(R\) complete extends to general \(R\) by pulling back from the \(p\)-adic completion of \(R\).
scientific article; zbMATH DE number 4156602
scientific article